02-09-2017, 03:36 PM
A central tool is the computation of the Fourier domain of an approximate digital radon transform. We introduce a very simple interpolation in the Fourier space that takes Cartesian samples and produces samples in a rectopolar grid, which is a pseudo-polar sampling set based on a geometry of concentric squares. Despite the harshness of our interpolation, the visual performance is surprisingly good. Our ridgelet transformation is applied to the radon transformation of a special supercomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our transformlet curvelet uses our ridgelet transform as a component step, and implements curvelet subbands using a bank of a / spl / grave filter filters. Our philosophy throughout is that transformations must be excessive, rather than being critically sampled. We apply these digital transformations to the destruction of some standard images embedded in white noise. In the trials presented here, the simple threshold of the curve coefficients is very competitive with wave-based "state of the art" techniques, including the threshold of deciduous or non-deciduous wave transformations and also including posterior average Bayesian methods in trees. On the other hand, curvelet reconstructions have a higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, a recovery of higher quality edges and weak linear and curvilinear traits. The existing theory for curvelet and ridgelet transformations suggests that these new approaches may outperform wavelet methods in certain image reconstruction problems. The empirical results presented here are in encouraging the agreement.